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Given a strictly increasing continuous function ϕ : R ≥ 0 ⟶ R ∪ { - ∞ } with lim t → ∞ ϕ (t) = ∞ , a function f : [ a , b ] ⟶ R ≥ 0 is said to be ϕ -concave if ϕ ∘ f is concave. When ϕ (t) = t p , p > 0 , this notion is that of p-concavity whereas for ϕ (t) = log (t) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set K ⊂ R n (of positive volume) w.r.t. some hyperplane H passing through its centroid is ϕ -concave, then one can find a sharp lower bound for the ratio vol (K -) / vol (K) , where K - denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for ϕ -concave functions (and involving the centroid) are shown. [ABSTRACT FROM AUTHOR]